3.626 \(\int \frac{\left (a+b x^4\right )^2}{x^2} \, dx\)

Optimal. Leaf size=28 \[ -\frac{a^2}{x}+\frac{2}{3} a b x^3+\frac{b^2 x^7}{7} \]

[Out]

-(a^2/x) + (2*a*b*x^3)/3 + (b^2*x^7)/7

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Rubi [A]  time = 0.0292538, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{a^2}{x}+\frac{2}{3} a b x^3+\frac{b^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^2/x^2,x]

[Out]

-(a^2/x) + (2*a*b*x^3)/3 + (b^2*x^7)/7

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Rubi in Sympy [A]  time = 5.01804, size = 22, normalized size = 0.79 \[ - \frac{a^{2}}{x} + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**2/x**2,x)

[Out]

-a**2/x + 2*a*b*x**3/3 + b**2*x**7/7

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Mathematica [A]  time = 0.00154136, size = 28, normalized size = 1. \[ -\frac{a^2}{x}+\frac{2}{3} a b x^3+\frac{b^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^2/x^2,x]

[Out]

-(a^2/x) + (2*a*b*x^3)/3 + (b^2*x^7)/7

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Maple [A]  time = 0.004, size = 25, normalized size = 0.9 \[ -{\frac{{a}^{2}}{x}}+{\frac{2\,ab{x}^{3}}{3}}+{\frac{{b}^{2}{x}^{7}}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^2/x^2,x)

[Out]

-a^2/x+2/3*a*b*x^3+1/7*b^2*x^7

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Maxima [A]  time = 1.43873, size = 32, normalized size = 1.14 \[ \frac{1}{7} \, b^{2} x^{7} + \frac{2}{3} \, a b x^{3} - \frac{a^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2/x^2,x, algorithm="maxima")

[Out]

1/7*b^2*x^7 + 2/3*a*b*x^3 - a^2/x

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Fricas [A]  time = 0.212032, size = 35, normalized size = 1.25 \[ \frac{3 \, b^{2} x^{8} + 14 \, a b x^{4} - 21 \, a^{2}}{21 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2/x^2,x, algorithm="fricas")

[Out]

1/21*(3*b^2*x^8 + 14*a*b*x^4 - 21*a^2)/x

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Sympy [A]  time = 1.00383, size = 22, normalized size = 0.79 \[ - \frac{a^{2}}{x} + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**2/x**2,x)

[Out]

-a**2/x + 2*a*b*x**3/3 + b**2*x**7/7

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GIAC/XCAS [A]  time = 0.233662, size = 32, normalized size = 1.14 \[ \frac{1}{7} \, b^{2} x^{7} + \frac{2}{3} \, a b x^{3} - \frac{a^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2/x^2,x, algorithm="giac")

[Out]

1/7*b^2*x^7 + 2/3*a*b*x^3 - a^2/x